Certain Classes Of Entire Functions Research Articles

A $ q $-Type $ k $-Lidstone series for entire functions

<abstract><p>In this paper, we consider the $ q $-type $ k $-Lidstone series. The series follows from expanding certain classes of entire functions in terms of Jackson $ q^{-1} $- derivatives at integers congruent to r modulo k, where $ k $ is a positive integer. We study the main properties of the fundamental polynomials that appear in the series expansion. We include a detailed study for the case $ k = 3 $ with some examples.</p></abstract>

Ax–Schanuel type theorems on functional transcendence via Nevanlinna theory

In this paper, we apply Nevanlinna theory to prove two Ax–Schanuel type theorems for functional transcendence when the original exponential map is replaced by other meromorphic functions. We give examples to show that these results are optimal. As a byproduct, we also show that analytic dependence implies algebraic dependence for certain classes of entire functions. Finally, some links to transcendental number theory and geometric Ax–Schanuel theorem will be discussed.

On a Certain Class of Entire Functions

In this paper, we examine the problem on the description in terms of the Laplace transform of functionals from the conjugate space for the Hilbert space of entire functions of n variables constructed by a convex function in ℂn, which depends on the modules of variables and grows at infinity faster than a‖z‖ for any a > 0.

The Saff–Varga Width Conjecture and Entire Functions with Simple Exponential Growth

We show that the partial sums of the power series for a certain class of entire functions possess scaling limits in various directions in the complex plane. In doing so, we obtain information about the zeros of the partial sums. We will only assume that these entire functions have a certain asymptotic behavior at infinity. With this information, we will partially verify for this class of functions a conjecture on the location of the zeros of their partial sums known as the Saff–Varga width conjecture. Numerical results and figures are included to illustrate the results obtained for several well-known functions including the Airy functions and the parabolic cylinder functions.

Real zeros of some classes of analytic functions determined by a majorant of infinite order

Certain classes of entire functions or of functions analytic in the unit disk are treated; they are defined in terms of a radial majorant λ that grows sufficiently fast. Under certain assumptions on λ, we describe the zero sets for such a class that lie on ℝ+ (respectively, on the segment [0, 1)). Bibliography: 9 titles.

Bessel harmonic analysis and approximation of functions on the half-line

We study problems of approximation of functions on in the metric of with power weight using generalized Bessel shifts. We prove analogues of direct Jackson theorems for the modulus of smoothness of arbitrary order defined in terms of generalized Bessel shifts. We establish the equivalence of the modulus of smoothness and the -functional. We define function spaces of Nikol'skii-Besov type and describe them in terms of best approximations. As a tool for approximation, we use a certain class of entire functions of exponential type. In this class, we prove analogues of Bernstein's inequality and others for the Bessel differential operator and its fractional powers. The main tool we use to solve these problems is Bessel harmonic analysis.

Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros

It is shown that new inequalities for certain classes of entire functions can be obtained by applying the Schwarz lemma and its generalizations to specially constructed Blaschke products. In particular, for entire functions of exponential type whose zeros lie in the closed lower half-plane, distortion theorems, including the two-point distortion theorem on the real axis, are proved. Similar results are established for polynomials with zeros in the closed unit disk. The classical theorems by Turan and Ankeny-Rivlin are refined. In addition, a theorem on the mutual disposition of the zeros and critical points of a polynomial is proved. Bibliography: 16 titles.

Padé approximants for entire functions with regularly decreasing Taylor coefficients

For a class of entire functions the asymptotic behaviour of the Hadamard determinants as and is described. This enables one to study the behaviour of parabolic sequences from Padé and Chebyshev tables for many individual entire functions. The central result of the paper is as follows: for some sequences in certain classes of entire functions (with regular Taylor coefficients) the Padé approximants , which provide the locally best possible rational approximations, converge to the given function uniformly on the compact set with asymptotically best rate.

Polynomial controllers for linear systems

In this paper, we study the problem of controlling a linear system with polynomial controls. We establish upper bounds of admissible polynomial controls in two special cases: systems with distinct eigenvalues and systems with all real eigenvalues. For certain other classes we show that the existence of such control of a minimum possible degree is equivalent to questions about the existence of multiple zeros for certain classes of entire functions.

Widths of certain classes of entire functions

The asymptotic behaviour of widths of classes of entire functions in the uniform metric on a compact set is studied. A logarithmic asymptotic formula is obtained which contains not only the natural parameters defining the growth at infinity of functions in the class under consideration, but also the capacity of the compact set. Under certain additional conditions a weak asymptotic formula is obtained. An example of the calculation of the strong asymptotics is presented.

On rational approximations of values of a certain class of entire functions

A sharp lower estimate is proved for the measure of irrationality of the values of -functions satisfying a system of linear differential equations of arbitrary order at a rational point.

Local uniform convergence and convergence of Julia sets

Let fn and f be entire functions and suppose that fn converges to f locally uniformly on C. Then if the Fatou set of f consists only of basins of attracting cycles or is empty, the Julia set of fn converges to that of f in the Hausdorff metric. We also show that expanding f implies the above assumption. Next we show that for each singular value c of f there exists a singular value c(n) of fn (for each sufficiently large n) converging to c. As an application, we propose a criterion which determines by the approximating sequence (fn)n=0infinity whether the Julia set of S is the whole sphere C for a certain class of entire functions.

Zeros of entire functions and characteristic determinants

We study certain classes of entire functions which arise as characterstics determinants of trace-class operators on a Hilber space. In particular, we show that the property of being as finitd oinear combination of commutatiors of Hilbert Schmidt operators can be characterized by a simple growth condtion on the charact.

On factorization of a certain class of entire functions

On factorization of a certain class of entire functions

On the univalence of the derivatives of a certain class of entire functions

On the univalence of the derivatives of a certain class of entire functions

On right and left factors of a certain class of entire functions

On right and left factors of a certain class of entire functions

Birkhoff interpolation of entire functions

We study the interpolation of certain classes of entire functions and their derivatives at the nonnegative integers, which, under certain conditions, reduces to the Birkoff interpolation of polynomials.

Solvability of partial differential equations of infinite order in certain classes of entire functions

In this paper it is shown that under conditions of applicability of the operator\(\mathfrak{L}y = \sum\nolimits_{k \geqslant 0} {a_k D^k y(x)}\) to the class [ρ,σ] ρ=(I,ρs), ρ2 <i, σ=(σ1, σ2), σ1, σ2<∞ the equation\(\mathfrak{L}\)y=f has a particular solution of this class vfe[ρ, σ]. The general form of a solution of the homogeneous equation\(\mathfrak{L}\)y=0 is established. The growth of a solution is investigated by means of a system of conjugate orders and a system of conjugate types. A solvability result is also obtained in the class\(E(T) = \mathop \cup \limits_{\sigma \in T} [\rho ,\sigma ]\), where T is a certain set in R + 2 depending on the operator\(\mathfrak{L}\).

An interpolation problem in certain classes of entire functions

An interpolation problem in certain classes of entire functions

Normal solvability of a differential operator on certain classes of entire functions

Normal solvability of a differential operator on certain classes of entire functions